Finite Group Databasebeta

← #26147294

Subgroups — #26147294

Order 12032.

3274 conjugacy classes of subgroups; only the 411 normal subgroups are shown.

normal subgroups only

This group has 3274 conjugacy classes of subgroups — too many to draw. Only its 411 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688979C212yesmaximal, complemented
2#20241637C216yesmaximal, complemented
3#20241629C216yesmaximal, complemented
4C22×#120×C47C211yesmaximal, characteristic
5C22×C22⋊Dic47C22112yes—
6C22×C22⋊Dic47C2214yescomplemented
7#15688979—28no—
8C2×C47⋊#95C22112yescomplemented
9C26×C47C2211yescharacteristic
10C23×D4×C47C2213yes—
11C2×#120×C47—28no—
12C2×#120×C47C418yescomplemented
13C22×C22⋊C4×C47C2213yes—
14C23×Dic47C2318yes—
15C2×C22⋊Dic47—224no—
16C2×C22⋊Dic47—424no—
17C2×C22⋊Dic47—412no—
18C47⋊#95—412no—
19C25×C47—412no—
20C25×C47C2×C414yes—
21C25×C47D418yes—
22C25×C47C2311yescharacteristic
23C25×C47C2313yes—
24C22×D4×C47—212no—
25C22×D4×C47—412no—
26C22×D4×C47C2×C4112yes—
27C23×C4×C47C2313yes—
28#120×C47—232no—
29C2×C22⋊C4×C47—212no—
30C2×C22⋊C4×C47C2×C4112yes—
31C24×C47—212no—
32C24×C47—224no—
33C24×C47—22no—
34C24×C47—212no—
35C24×C47—424no—
36C24×C47—448no—
37C24×C47—448no—
38C24×C47—416no—
39C24×C47C22⋊C4116yescomplemented
40C24×C47C22×C412yes—
41C24×C47C22×C416yes—
42C24×C47C2×D4112yes—
43C24×C47C2411yescharacteristic
44C2×D4×C47—224no—
45C2×D4×C47—424no—
46C2×D4×C47—448no—
47C22×C4×C47—26no—
48C22×C4×C47—212no—
49C22×C4×C47C22×C416yes—
50C22×Dic47—28no—
51C22×Dic47—224no—
52C22×Dic47—224no—
53C22⋊C4×C47—224no—
54C22⋊Dic47—448no—
55C22⋊Dic47—416no—
56C23×C47—212no—
57C23×C47—224no—
58C23×C47—224no—
59C23×C47—224no—
60C23×C47—224no—
61C23×C47—22no—
62C23×C47—212no—
63C23×C47—448no—
64C23×C47—464no—
65C23×C47—4128no—
66C23×C47—436no—
67C23×C47—412no—
68C23×C47—412no—
69C23×C47—412no—
70C23×C47—44no—
71C23×C47C2×C22⋊C4112yes—
72C23×C47C23×C411yescharacteristic
73C23×C47C22×D412yes—
74C23×C47H(5,𝔽2)112yes—
75D4×C47—448no—
76C2×C4×C47—26no—
77C2×C4×C47—212no—
78C2×C4×C47—412no—
79C2×Dic47—424no—
80C2×Dic47—424no—
81C2×Dic47—48no—
82#14657—471nomaximal, Sylow
83C22×C47—212no—
84C22×C47—224no—
85C22×C47—464no—
86C22×C47—424no—
87C22×C47—448no—
88C22×C47—46no—
89C22×C47—48no—
90C22×C47—88no—
91C22×C47C22×C22⋊C411yescharacteristic
92C22×C47#51816yes—
93C22×C47#534112yes—
94C22×C47#560112yes—
95C22×C47H(3,𝔽4)14yes—
96C4×C47—46no—
97Dic47—88no—
98C22×#120D4711yescharacteristic
99C2×#379—472no—
100#1440—476no—
101#1438—476no—
102C2×C47—412no—
103C2×C47—82no—
104C2×C47#230813yes—
105C2×C47#237514yes—
106C2×C47#281218yes—
107C26C2×D4711yescharacteristic
108C23×D4C2×D4713yes—
109C2×#120—28no—
110C2×#120Dic4718yescomplemented
111C22×C22⋊C4—4712no—
112C22×C22⋊C4—474no—
113C22×C22⋊C4C2×D4713yes—
114#379—948no—
115C2×#95—4712no—
116C47#1465711yescharacteristic, complemented, Sylow
117C25—412no—
118C25C2×Dic4714yes—
119C25C22⋊D4718yes—
120C25C22×D4711yescharacteristic
121C25C22×D4713yes—
122C22×D4—212no—
123C22×D4—412no—
124C22×D4C2×Dic47112yes—
125C23×C4—478no—
126C23×C4C22×D4713yes—
127#120—232no—
128C2×C22⋊C4—212no—
129C2×C22⋊C4—9424no—
130C2×C22⋊C4—18824no—
131C2×C22⋊C4—18812no—
132C2×C22⋊C4C2×Dic47112yes—
133#95—18812no—
134C24—224no—
135C24—212no—
136C24—22no—
137C24—212no—
138C24—424no—
139C24—448no—
140C24—448no—
141C24—416no—
142C24C22⋊Dic47116yescomplemented
143C24C22×Dic4712yes—
144C24C22×Dic4716yes—
145C24C2×C22⋊D47112yes—
146C24C23×D4711yescharacteristic
147C2×D4—224no—
148C2×D4—424no—
149C2×D4—448no—
150C22×C4—26no—
151C22×C4—212no—
152C22×C4—948no—
153C22×C4—9424no—
154C22×C4—9424no—
155C22×C4C22×Dic4716yes—
156C22⋊C4—224no—
157C22⋊C4—18848no—
158C22⋊C4—18816no—
159C23—212no—
160C23—224no—
161C23—224no—
162C23—224no—
163C23—224no—
164C23—22no—
165C23—212no—
166C23—448no—
167C23—464no—
168C23—4128no—
169C23—436no—
170C23—412no—
171C23—412no—
172C23—44no—
173C23—412no—
174C23C2×C22⋊Dic47112yes—
175C23C23×Dic4711yescharacteristic
176C23C22×C22⋊D4712yes—
177C23#13094069112yes—
178D4—448no—
179C2×C4—26no—
180C2×C4—212no—
181C2×C4—412no—
182C2×C4—18824no—
183C2×C4—1888no—
184C2×C4—18824no—
185C22—212no—
186C22—224no—
187C22—464no—
188C22—424no—
189C22—448no—
190C22—48no—
191C22—46no—
192C22—88no—
193C22C22×C22⋊Dic4711yescharacteristic
194C22#1568959816yes—
195C22#15689682112yes—
196C22#15689862112yes—
197C22#1568986614yes—
198C4—46no—
199C4—3768no—
200C2—412no—
201C2—82no—
202C2#2024596613yes—
203C2#2024652914yes—
204C2#2025138318yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
Fitting, prime radicalC22×#120×C47C2
socle, cosocleC24×C47C24
hypercenterC22×#120D47
hypocenterC47#14657
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×#120

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#15688979C2semidirect
2. #20241637C2semidirect
3. #20241629C2semidirect
6. C22×C22⋊Dic47C22semidirect
8. C2×C47⋊#95C22semidirect
12. C2×#120×C47C4semidirect
39. C24×C47C22⋊C4semidirect
110. C2×#120Dic47semidirect
116. C47#14657semidirect
142. C24C22⋊Dic47semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#156889792
#202416372
#202416292
C22×#120×C472
#1465747